"A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." – Paul Halmos

By the formula $All = exp(Connected)$ (http://mathoverflow.net/a/215053/1015), $a_n$ counts the number of connected (a.k.a. indecomposable) permutations on $n$ elements, i.e., permutations which do not fix any interval $[1..p]$ for $p < n$. Perhaps more surprisingly, $a_n$ also counts the number of rooted oriented hypermaps with $n$ edges. (A rooted oriented hypermap is a hypergraph embedded in a compact oriented surface without boundary, equipped with a choice of edge. A bijection between indecomposable permutations and rooted oriented hypermaps was given by Patrice Ossona de Mendez and Pierre Rosenstiehl. It restricts to a bijection between indecomposable involutions and rooted oriented maps, which is nicely described in their short paper, "Encoding pointed maps by double occurrence words".)

Of course you’re right, I should have said indecomposable permutations on $n+1$ elements. From your hint in the next post, though, I suspect you were going for the second answer, rooted oriented hypermaps on $n$ edges, viewed as pairs of permutations acting transitively on a set. Is that right?

I’m making a mistake somewhere below but can’t figure out where I’m going wrong.

Since exponential generating functions (EGFs) are better behaved than ordinary generating functions (OGFs), let’s pretend that is an EGF counting pairs of permutations on a set of elements. That is equivalent to being given two sets and of elements each and counting pairs of bijections and . This in turn is equivalent to counting the number of bipartite digraphs where and are the partite sets and every vertex has exactly one incoming edge and exactly one outgoing edge. Such digraphs are unordered sets of connected bipartite digraphs where and are the partite sets and every vertex has exactly one incoming edge and exactly one outgoing edge. So, , regarded as an EGF, counts the latter. The coefficients are related to via $b_n = a_n(n-1)!$. So, ought to count the number of quotient set by obtained by identifying the connected bipartite digraphs above whenever they are related by a cyclic permutation of the elements of .

This unfortunately doesn’t agree with the first few instances given in the post.